# Point Estimation Parameters

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Category: Education

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### 7-1 Introduction :

7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These methods utilize the information contained in a sample from the population in drawing conclusions. Statistical inference may be divided into two major areas: Parameter estimation Hypothesis testing

### 7-1 Introduction :

Definition 7-1 Introduction

7-1 Introduction

7-1 Introduction

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.1 Unbiased Estimators Definition

### 7-2 General Concepts of Point Estimation :

Example 7-1 7-2 General Concepts of Point Estimation

### 7-2 General Concepts of Point Estimation :

Example 7-1 (continued) 7-2 General Concepts of Point Estimation

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.3 Variance of a Point Estimator Definition Figure 7-1 The sampling distributions of two unbiased estimators

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.3 Variance of a Point Estimator Theorem 7-1

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.4 Standard Error: Reporting a Point Estimate Definition

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.4 Standard Error: Reporting a Point Estimate

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation Example 7-2

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation Example 7-2 (continued)

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator Definition

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator

### 7-2 General Concepts of Point Estimation :

7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator Figure 7-2 A biased estimator that has smaller variance than the unbiased estimator

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Definition Definition

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-4

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation 7-3.2 Method of Maximum Likelihood Definition

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-6

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-6 (continued)

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Figure 7-3 Log likelihood for the exponential distribution, using the failure time data. (a) Log likelihood with n = 8 (original data). (b) Log likelihood if n = 8, 20, and 40.

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-9

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-9 (continued)

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Properties of the Maximum Likelihood Estimator

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation The Invariance Property

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-10

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Complications in Using Maximum Likelihood Estimation It is not always easy to maximize the likelihood function because the equation(s) obtained from dL()/d = 0 may be difficult to solve. It may not always be possible to use calculus methods directly to determine the maximum of L().

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Example 7-11

### 7-3 Methods of Point Estimation :

7-3 Methods of Point Estimation Figure 7-4 The likelihood function for the uniform distribution in Example 7-11.

### 7-4 Sampling Distributions :

7-4 Sampling Distributions Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population. Definition

### 7-5 Sampling Distributions of Means :

7-5 Sampling Distributions of Means Theorem 7-2: The Central Limit Theorem

### 7-5 Sampling Distributions of Means :

7-5 Sampling Distributions of Means Figure 7-6 Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter (1978).]

Example 7-13

### 7-5 Sampling Distributions of Means :

7-5 Sampling Distributions of Means Figure 7-7 Probability for Example 7-13.

### 7-5 Sampling Distributions of Means :

7-5 Sampling Distributions of Means Definition